Those two small wires on the battery are a good place to start.
The factory did not put them there
If the engine cranks without the key in place it has a major wiring fault in the cranking circuit.
So start by getting the wiring diagram for your mower
Hustler put them in both the service manuals and parts books which are a free download
For us to get the right one we need the full mower details from the ID tag, model & serial numbers.
Most Hustlers use a cranking relay which powers the starting solenoid
It could be bad or the OP could have installed the wrong one
If the mower had a kawasaki then it may have a piggy back starting solenoid
If it has a Briggs then it will have a remote solenoid so we need more information to proceed